Ingham

Albert Edward Ingham

Born: 3 April 1900 in Northampton, England
Died: 6 Sept 1967 in Chamonix, France

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Ingham won a scholarship to Trinity College Cambridge in December 1917, spent a few months in the army, then in January 1919 he began his studies. An outstanding undergraduate career saw him win a Smith's prize and the highest honours. In 1922 he was elected to a fellowship at Trinity for a dissertation on the zeta function and his next four years were occupied only with research, a few months of which were spent at Göttingen. During this time Ingham was greatly influenced by Littlewood who would give him the advice to
work at a hard problem: you may not solve it but you'll solve another one.
In 1926 Ingham was appointed to Leeds but 4 years later returned to Cambridge, on the death of Ramsey, and remained there for the rest of his life. He was elected a Fellow of the Royal Society in 1945 and became a Reader in Mathematical Analysis in 1953.

His book On the distribution of prime numbers published in 1932 was his only book and it is a classic. Many of the ideas here, as in other work of Ingham's, came from the joint work undertaken by Harald Bohr and Littlewood.

Ingham's work was on the Riemann zeta function, the theory of numbers, the theory of series and Tauberian theorems.

He generalised work on the prime number theorem of Hadamard and Vallée Poussin. The result for which Ingham is best known, however, relates to pn+1 - pn where pn denotes the nth prime. It was proved by Hoheisel in 1930 that there is a constant k such that

pn+1 - pn < pn^k
for all sufficiently large n. In On the difference between two consecutive primes (1937) Ingham proved that the result holds for k = 5/8.

Pólya, in 1919, made the following conjecture. Suppose lambda(n) = 1 if n has an even number of prime factors, -1 if n has an odd number of prime factors (counting multiplicities) then let L(x) be the sum of lambda(n) over all positive integers less than n. The conjecture is that L(x) lt or equals 0. Ingham, in 1942, was able to find an ingenious method to show how a counterexample could be constructed. It still required computing power to find the counterexample and, using Ingham's method, a counterexample was found by R S Lehman in 1960 when he showed that L(906180359) = 1.

Some of Ingham's work on number theory was carried further by Linnik.

Ingham also worked on Tauberian theorems. He proved results suggested by Wiener, and he applied methods first developed by Wiener.

Ingham led a life of great simplicity. Burkill describes it in these terms:

It did not occur to him to want a car or a radio, let alone a television set. For forty years he used a Sunbeam bicycle that he had won as a school prize. He was an expert photographer: he developed his own colour films and did everything from first principles. He was a good cricketer...who would have been of minor county class if he had been able to give the time.
He died while on a walking holiday in the mountains. He and his wife had taken this type of holiday every summer for many years.

References:

  1. J C Burkill, Albert Edward Ingham, Bull. London Math. Soc. 1 (1969), 109-124. 
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